Elementary Geometry For College Students, 7e
7th Edition
ISBN: 9781337614085
Author: Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher: Cengage,
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![**Problem Statement: Volume of a Regular Triangular Pyramid**
**Objective:** Determine the volume of a regular triangular pyramid given certain measurements.
**Given:**
- The length of one side of the base is 12 cm.
- The height of the pyramid is 4 cm.
**Solution:**
To find the volume of a regular triangular pyramid, we use the formula:
\[ \text{Volume} = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
1. **Calculate the area of the base:**
Since the base is a regular triangle (equilateral triangle), the area \( A \) can be calculated using the formula:
\[ A = \frac{\sqrt{3}}{4} \times (\text{side length})^2 \]
Given the side length of the base is 12 cm, we plug it into the formula:
\[ A = \frac{\sqrt{3}}{4} \times (12 \, \text{cm})^2 \]
\[ A = \frac{\sqrt{3}}{4} \times 144 \, \text{cm}^2 \]
\[ A = 36\sqrt{3} \, \text{cm}^2 \]
2. **Substitute the base area and height into the volume formula:**
\[ \text{Volume} = \frac{1}{3} \times 36\sqrt{3} \, \text{cm}^2 \times 4 \, \text{cm} \]
\[ \text{Volume} = \frac{1}{3} \times 144\sqrt{3} \, \text{cm}^3 \]
\[ \text{Volume} = 48\sqrt{3} \, \text{cm}^3 \]
**Conclusion:**
The volume of the regular triangular pyramid is \( 48\sqrt{3} \, \text{cm}^3 \).](https://content.bartleby.com/qna-images/question/4895642d-adca-4243-b0c4-c741dfdd726d/808a884a-8452-4fd6-b1df-c30b1f3189e2/74rz2js_processed.jpeg)
Transcribed Image Text:**Problem Statement: Volume of a Regular Triangular Pyramid**
**Objective:** Determine the volume of a regular triangular pyramid given certain measurements.
**Given:**
- The length of one side of the base is 12 cm.
- The height of the pyramid is 4 cm.
**Solution:**
To find the volume of a regular triangular pyramid, we use the formula:
\[ \text{Volume} = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
1. **Calculate the area of the base:**
Since the base is a regular triangle (equilateral triangle), the area \( A \) can be calculated using the formula:
\[ A = \frac{\sqrt{3}}{4} \times (\text{side length})^2 \]
Given the side length of the base is 12 cm, we plug it into the formula:
\[ A = \frac{\sqrt{3}}{4} \times (12 \, \text{cm})^2 \]
\[ A = \frac{\sqrt{3}}{4} \times 144 \, \text{cm}^2 \]
\[ A = 36\sqrt{3} \, \text{cm}^2 \]
2. **Substitute the base area and height into the volume formula:**
\[ \text{Volume} = \frac{1}{3} \times 36\sqrt{3} \, \text{cm}^2 \times 4 \, \text{cm} \]
\[ \text{Volume} = \frac{1}{3} \times 144\sqrt{3} \, \text{cm}^3 \]
\[ \text{Volume} = 48\sqrt{3} \, \text{cm}^3 \]
**Conclusion:**
The volume of the regular triangular pyramid is \( 48\sqrt{3} \, \text{cm}^3 \).
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